I have this problem for my Calc class and, though I've done it manually and through my calculator, WeBWorK keeps telling me I'm wrong. I hoped someone could tell me if I'm interpreting it incorrectly or maybe just doing the wrong calculation.

A particle that moves along a straight line has velocity

[math]v(t) = t^2 e^{-2 t}[/math]

meters per second after t seconds. How many meters will it travel during the first t seconds?

To me, this means that I must find

[math]\int t^2 e^{-2 t} \ dt[/math]

According to everything I've tried, this equals [math]e^{-2t}(-t^{2}/2-t/2-1/4)[/math]

What am I doing wrong here?

You did not specify the start and endpoint of the integration. You must integrate from 0 to t. The function you give is a primitive of t²*exp(-2t), but it is not the covered distance at time t.

 

Let's call this primitive F(t). Then the solution to your problem is F(t) - F(0). Your answer is almost correct, you derived a correct primitive. The answer for the covered distance misses a constant term.

 

Another way to see that your answer is not complete is to fill in the value t = 0 in your answer. The result also should be zero, but it isn't. At time 0, no distance is covered yet.

Yes, really you ought to finding the integral

 

[math]\int_0^t v(x)dx[/math]

Of course! I'd actually tried that before, but I had made a mistake integrating at that point. Thanks for the help! This has been bugging me.